Even measuring the leading three principal components directly, which are optimal for linear reconstruction, cannot always reveal the state of the shock-mixing layer flow.
The black vertical lines in Figure REF  indicate the phases of two distinct states for which the leading three principal components agree, yet the fourth differs.
One may wonder whether the fact that the third and fourth principal components are one-to-one with the state can be leveraged for sensor placement.
Even our attempt to place three maximum likelihood D-optimal sensors using the convex optimization approach of [1]} to reconstruct the third and fourth principal components fails to produce measurement that can recover the phase of the flow as seen in Figure REF .
